Using Mathematical Structure - Pre Algebra

1. What is the value of $\frac{3}{4} \times 17 \times \frac{4}{3}$?

   17. The reciprocals $\frac{3}{4}$ and $\frac{4}{3}$ multiply to 1.

2. Pairing numbers that make a round sum, what is $47 + 38 + 53$?

   138, since $(47 + 53) + 38 = 100 + 38$.

3. What is the value of $\frac{36 \times 15}{12 \times 5}$?

   9. Simplify the factors first: $\frac{36}{12} = 3$ and $\frac{15}{5} = 3$.

4. What is the smallest possible value of $(x - 5)^2 + 3$?

   3. A square is never negative, so the least value occurs when $(x - 5)^2 = 0$.

5. What is the value of $2^5 \times 5^5$?

   100,000. Pairing the factors gives $(2 \times 5)^5 = 10^5$.

6. What is the value of $4^3 \times 25^3$?

   1,000,000. Pairing the factors gives $(4 \times 25)^3 = 100^3$.

7. Treating $(x + 1)$ as a single quantity, solve $3(x + 1) = 12$.

   $x = 3$. First $x + 1 = 4$, then $x = 3$.

8. Using $99 = 100 - 1$, what is $99 \times 7$?

   693, since $100 \times 7 - 1 \times 7 = 700 - 7$.

9. Thinking of $0.25$ as $\frac{1}{4}$, what is $0.25 \times 48$?

   12, since $\frac{1}{4}$ of 48 is 12.

10. Regrouping to make a friendly product, what is $25 \times 36 \times 4$?

    3,600, since $(25 \times 4) \times 36 = 100 \times 36$.

11. Pairing first and last terms, what is $1 + 2 + 3 + \cdots + 100$?

    5,050. There are 50 pairs ($1 + 100$, $2 + 99$, and so on), each adding to 101.

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